An Agent-based model to simulate the transmission of SARS-CoV-2 in meat processing facilities

SACADA Project

April 2021 - September 2022

1 Introduction

1.1 Background

This modelling study aimed to…

The Agent Based Model considered three sources of contamination: the exposition to SARS-CoV-2 througout working activities (aerosol and/or fomites), the contamination throught the exposition outside the processing plant for employees having shared activities in community (transport and/or living).

1.2 Objectives

In order to consider the first source of contamination (in the factory), the durations of the stages of infections and the Viral load (viral RNA concentration) exhaled by infected people have to be quantified. For assessing the possibility of transmission in community the secondary attack rate have to be quantified.

The different waves of Sars-CoV2 outbreaks that have occurred show that each Variants of Concern, i.e. VoC (alpha, beta, delta and omicron SARS-CoV-2 VoCs) have their own characteistics, e.g. different viral load, or rates of reinfection (Karim & Karim, 2021).

This document aims to identify the parameters of interest for building the Agent Based Model of the ANR Sacada project.

2 Agent-based modeling

The ABM model is presented in this study according to the Overview-Design-Details (ODD) protocol as recommended by Grimm et al. (2010).

2.1 Overview

The outline of the agent-based modelling workflow developed in this study is schematized in Figure 2.1.

Figure 2.1: Outline of the Agent-based model to simulate the transmission of SARS-CoV-2 in meat processing facilities

• Purpose and patterns

The developed model was aimed to predict …

• States variables and scales - Parameters overview

Several parameters were included in the model. On the one hand, some parameters… On the other hand, based on multiple industrial and epidemiological investigations, other parameters were included to describe the behavior of the model in different scenarios. These scenarios can be defined by ch as well as the behaviors of the agents

• Process overview and scheduling

The model was divided in four separate modules describing different agents evolving in the processing plant including (i) the workers, (ii) the droplets in the air, (iii) the inert surfaces and (iv) the food products, respectively. As schematized in Figure 2.1, different possible interactions between these agents were described, such as the transmission of the droplets probably containing viral particles between infectious workers and the air through their respiratory activities (exhalation and inhalation), the sedimentation of these droplets from the air onto inert surfaces and/or food portions as well as the transfers between surfaces and food, etc. Based on prior data or knowledge, spatiotemporal rules were defined to simulate the interactions between modules and the properties of every agents at successive time points distanced by a predefined short time step (minutes). These step-by-step simulations were performed afterwards for longer periods (days, weeks, etc.). At the end of these periods, balances were calculated for different properties of the agents, e.g. the total number of aerosolized or fallen droplets, the number of viral gene copies inhaled by each worker, etc. Based on these balances, several contamination indicators were estimated such as the total number of infected workers, the number of contaminated food portions and surfaces in the plant, etc. These estimated indicators could be compared with collected epidemiological data for further model validation purposes.

2.2 Design concept

2.2.1 Basic principles

The different agents integrated into the model for describing the functioning of the meat processing plant are illustrated in Figure 2.2. The model was implemented in the way that end-users can include by themselves as input data the different characteristics of the plant such as its dimensions, structures, objects, etc. For instance, as shown in Figure 2.2, the illustrated plant is made up of several rooms / spaces including an entry hall for workers, an office, a cooling area for food storage, a large area for meat cutting activities with two conveyor belts for food transfers and a packaging machine, etc. Different types of workers were considered in the model depending on their respective activities, including the cutters, the logistic workers as well as the ones with transverse activities. The number of workers of each type present in the plant and their respective working rhythm and timetable condition then the daily meat processing flow (meat quantity, processing duration) of the overall plant. This processing flow was described as successive transformation steps of the meat products: the carcasses/half-carcasses arriving in the plant through the gate are transported to the conveyors by logistic workers, become smaller meat portions after different cutting steps carried out by cutters around the conveyors and eventually packed and stored in the cooling area. Other workers with transverse activities such as manager, administrative or cleaning workers can also be present in the plant at different moments depending on their respective timetable. Based on the overall timetable of the plant, one can define rules for generating predefined or random movements of every agents (workers, carcasses, meat cuts…) between different locations in the plant at different time points. Determining these movements over time will allow then to track the possible transmission of the virus introduced by one or several infectious workers in the plant.

Figure 2.2: Illustration : Modeling the meat processing plant

The transmission of the virus from one or several infectious workers to other susceptible workers, surfaces and food products are schematized in the Figure 2.3. In this study, the duration between the different successive infection phases were taken into account from new infection to recovery with/without symptoms development. The different infectious workers can exhale into the air, through their respiratory activities or symptoms such as breathing, talking, sneezing or coughing, different amounts of droplets containing or not viral particles due to variability between workers and depending on their behavior (mask wearing). These droplets of various sizes can be aerosolized and/or fall afterwards onto inert surfaces and food products depending on the characteristics of the ventilation system in each space of the plant. Some high-diameter droplets sent out by the infectious workers (e.g. in cases of sneezing or coughing) can also fall directly onto surfaces without being aerosolized and/or projected towards nearby workers. The total viral loads on different surfaces and/or food products (meat cuts) are also enumerated; they are considered as contaminated if the viral loads exceed a predefined threshold value. The non-infected workers (susceptible workers) can inhale more or less of droplets depending on their positions in the plant at different moments. The cumulative viral doses inhaled by each susceptible worker at the end of the day can then probably trigger an infection following a dose-response relationship. At last, in addition of these possible infections inside the plant, the model included also other contamination sources outside the plant: the susceptible workers can get infected due to the outside epidemic situation and or by community activities such as co-living with infectious workers.

Figure 2.3: Modeling the transmission of SARS-CoV-2 between different agents

2.2.2 Prediction

The parameters

2.2.3 Stochasticity

The stochasticity

2.3 Details

2.3.1 Sub-models - Modules

2.3.1.1 Food processing plant

2.3.1.1.1 Structures

The plan of the multi-room food processing plant can be taken into account in the model. Based on industrial investigations, seven principal rooms (spaces) were included herein as an example to define the plant structure, including the Cutting room (the main space of the plant), Entry hall (for workers), Arrival gate (for food arrival into the cutting room), Cooling area (for food storage), Waste area, W.C., and Office (illustration in Figure 2.2).

The different characteristics of each room are included as inputs of the model:

• dimensions of the plant and dimensions of the different rooms, expressed in m (the length and width on two axes, denoted X and Y axes, respectively, and height on a 3rd axis, denoted Z axis);
• positions of the room in the plant;
• presence and relative positions of internal/external door for each room;
• ventilation specifications: several air conditioning types can be indicated such as Controlled Mechanical Ventilation (CMV), Direct Expansion Coil (DEC) or Air Handing Unit (AHU).

The above characteristics were defined in the model as model parameters based on industrial investigations or previous works. For example, each room needs to be positioned on the entire workshop grid using a numerical value between 0 and 1 indicating the relative position over the axes. The cutting room is the main build by subtracting all other rooms. The position of the internal and external doors can also be mentioned using the predefined values “top”, “bottom”, “left” or “right”.

2.3.1.1.2 Objects

The different objects or equipment can be included inside the plant such as conveyors, peeler, etc. In the same ways as for the different rooms, their dimensions and their relative position alongside the X and Y axes are taken into account as model inputs. As illustrated in Figure 2.2, two types of objects were included : two conveyor belts allowing to transport the different carcasses or food portions from one point to another and an equipment that can be shared by several operators (e.g.: a peeler, frequently used in meat cutting plant).

Table 2.1: Set of parameters for charateryzing the different objects in the plant.

Parameters	name of the object	length of the object	width of the object	Position of regarding the X axis of the plant	Position regarding the Y axis of the plant
Label in ABM script	label	dim.X	dim.Y	pos.X	pos.Y
unit		meter	meter	0 to 1 (0: left, 0.5: middle, 1:right)	0 to 1 (0: bottom, 0.5: middle, 1:top)
cvy	Conveyor1	13	3	0.2	0.5
cvy2	Conveyor2	18	3	0.5	0.5
epmA	Equipment 1	2	2	0.85	0.7

2.3.1.2 Workers

2.3.1.2.1 Type of workers

Different principal types of workers were taken into account in the model depending on the their daily tasks inside the plant. According to industrial investigations and expert knowledge, six types of workers were included as following (Figure 2.2):

• logistic1 : the workers in charge of bringing carcasses (or half-carcasses), arriving in the cutting room through the arrival gate, to the begin of the conveyor for cutting.
• cutter1 : positioned around the first section of the conveyor, denoted conveyor1, in charge of deboning, trimming and cutting carcasses into large pieces;
• cutter2 : workers in charge of cutting larges pieces from cutter1 into smaller food portions;
• logistic2 : the workers in charge of transferring food portions from the cutting tables to the packaging or supplementary treatment equipment and storing packed products in the cooling area;
• transverse1: administrative workers such as manager, chief or maintenance staff who mainly stay out of the cutting room during the work day;
• transverse2: people from the cleaning team.

The total number of employees is defined as shown is Table 2.2.

Table 2.2: Set of parameters used for charateryzing the worker of the plant.

Parameters	Label in ABM script	Value	Unit	Uncertainty range	References
Total number of worker	Nworkers	100.000	N/A	[??-??]	[interviews]
Number of regular worker	Nworkers_ext	15.000	N/A	[1-4]	[interviews]
Number of temporary worker		NA
Number of subcontract worker		NA
Overall proportion for different types of workers
Cutter	cutter1	0.360	ratio (0 to 1)	[5-11]	(Mallet et al., 2021)
Cutter	cutter2	0.470	ratio (0 to 1)	[5-11]	(Mallet et al., 2021)
Logistic	logistic1	0.040	ratio (0 to 1)		(Gunther et al., 2020)
Logistic	logistic2	0.040	ratio (0 to 1)		(Gunther et al., 2020)
Transverse	transverse1	0.045	ratio (0 to 1)	[0.18-0.475]	(Mallet et al., 2021)
Transverse	transverse2	0.045	ratio (0 to 1)		(Mallet et al., 2021)

The workers were split afterwards into two teams, denoted “team A” and “team B” according to their working shift (morning or afternoon) and to their weekly schedule. In the model, this team dividing was included by the parameters pTeam (value between 0 and 1 representing the proportion of each team) and set by default at 0.5 (equal distribution between the two teams A and B). In each team, the types of workers (cutters, transverse and logistic workers) were proportionally distributed as described above. The two teams A and B alternatively change their shift (morning/afternoon) from one week to another. Finally, the workers change the team (between the teams A and B) from one week to another using a probability defined as model parameter, denoted pChangeTeam, (value between 0 and 1), arbitrarily set as 0.05.

2.3.1.2.2 Infection phases

The duration of different infections phases were taken into account in the model and are schematized in Figure 2.4). The uncertainty ranges of the different values have been identified for sensitivity analysis of the hypothesis of the model (Table 2.3).

Once infected, individuals enter firstly into a presymptomatic phase including (i) a latent period during which no virus can be detected and (ii) individuals become infectious before symptom onset (Lau & Leung, 2020). The duration of the latent period has been estimated to 3.3 days with 95% CI [0.2-7.9] (Zhao et al., 2021). The infectious period of the presymtomatic period has a length duration of 2 days (with an uncertainty range between 1 and 4 days) (Byrne et al., 2020). Considering the data available for the delta VoC (Grant et al., 2021; Kang et al., 2021), the presymptomatic period duration was considered to be equivalent for every VoCs. At the end of the presymptomatic period, a proportion of the individual will develop symptoms. Several studies are available for estimating this proportion (Alene et al., 2021; J. He, Guo, Mao, & Zhang, 2021; Ma et al., 2021; Sah et al., 2021). Alene et al. (2021) estimated the pooled proportion of asymptomatic COVID-19 cases throughout the course of infection was 25% (with 95%CI: 16%-38%). Ma et al. (2021) estimated that the pooled percentage of asymptomatic infections among the confirmed population was 40.50% (with 95%CI: 33.50%-47.50%). Sah et al. (2021) estimated that percentage at 35.1% (with 95% CI: 30.7-39.9%). In our study, the mean of these three percentages has been used as model parameter, pAsymptom, i.e. 33.5%. The range of possible values was also derived from the 95%CI of these studies and set between 16% and 47.5% in our study.

For the sub-population who develops symptoms, the duration of their symptomatic period has been found uncertain according to studies (Byrne et al., 2020). The duration of the different symptoms differ according to their nature (Santos et al., 2021). From the onset of symptoms to recovery, a period of 13 days can be considered with an uncertainty range comprise between 11 and 16 days (Byrne et al., 2020). During that period, virus can be detected but it as estimated that infectiousness becomes usually low after 7 days (X. He et al., 2020). The uncertainty range for the infectiousness period can be estimated from Byrne et al. (2020). They stated that virus detection overestimates the infectious period on average by 2-6 days. Kampen et al. (2021) provide another estimation of the uncertainty range of infectiousness period. They considered the duration between 5 and 11 days. The value of 8 days (Kampen et al., 2021) was used in the model .

Figure 2.4: Representation of the stage of infection for SARS-CoV2

Table 2.3: Set of parameters used for charateryzing the durations of the SARS-CoV2 infection.

Parameters	Label in ABM script	Value	Unit	Uncertainty range	References
Latent phase of presymptomatic period	InfectiousBeginDay	3	days	[0-8]	(Zhao et al., 2021)
Infectious phase of presymtomatic period	xx	2	days	[1-4]	(Byrne et al., 2020)
Duration of the presymtomatic period	PossibleSymptomeBeginDay	InfectiousBeginDay+XX	days
Infectious phase after presymptomatic period for symptomatic people	yy	8	days	[5-11]	(Byrne et al., 2020; Kampen et al., 2021)
Infectious phase after presymptomatic period for asymptomatic people	zz	8	days	[5-11]	(Byrne et al., 2020)
End of the infectious period for symtomactic	NonInfectiousBeginDay	PossibleSymptomBegin+yy	days
End of the infectious period for asymtomactic	NonInfectiousBeginDayAS (Du)	PossibleSymptomBegin+zz	days
Proportion of asymptomatic		0.335		[0.18-0.475]	(Alene et al., 2021; Ma et al., 2021; Sah et al., 2021)

2.3.1.2.3 Individual variability

Among the infection phases described previously, the model considered the viral load emitted by the different workers only during their infectiousness period. Viral loads were considered to be similar between asymptomatic and symptomatic persons according to a study of Walsh et al. (2020) and assumed to be constant during the whole infectiousness period. The Agent-Based model takes into account the inter-individual variability of virus load in sputum and the VoC. Data obtained by Jones et al. (2021) on presymptomatic, asymptomatic, and mildly symptomatic (PAMS) cases were used to estimated this inter-individual variability. The latter corresponds to the alpha variant (that is B.1.1.7). The viral load of the alpha variant was found to be 1 \(\log_{10}\) higher than the original SARS-CoV-2 strain (Teyssou et al., 2021).

Figure 2.5: Fitting of a triangular distribution to virus load measurements (expressed in log10 copy/ml) on data obtained by Jones et al (2021).

Table 2.4: Fitted triangular distribution for charaterizing the individual variability of viral loads in log10 (copy/ml) in stupum.

VoC	Min	Mode	Max	Shift compared to delta	Reference
delta	1.6	8.6	10.4	NA	(Jones et al., 2021)
original	0.6	7.6	9.4	-1.0	(Teyssou et al., 2021)
alpha	1.1	8.1	9.9	-0.5	(Luo et al., 2021; Teyssou et al., 2021; Wang et al., 2021)
omicron	1.6	8.6	10.4	0.0	(Lyngse et al., 2021)

The parameters of the triangular distribution used to describe variability of viral load are respectively, min=1.6, mode=8.6 and max=10.4.

The delta VoC is associated with higher viral load (Luo et al., 2021; Teyssou et al., 2021; Wang et al., 2021). The concentration of delta VoC was found to be 2 times more important than for alpha (Teyssou et al., 2021). Other differences observed in Ct values suggest that the difference could be higher (Luo et al., 2021; Wang et al., 2021). Therefore, the distribution of viral load was considered to be shifted by 0.5 \(\log_{10}\) from delta with an uncertainty range between 0.3 and 1 \(\log_{10}\). Finally, the level of viral load for omicron VoC was found to be similar to delta (Lyngse et al., 2021).

2.3.1.2.4 Secondary attack rate

The secondary attack rate (SAR) parameter used to estimate the probability of contraction SARS-CoV-2 for the workers having community activities outside the plant such as co-living. The SAR of the original and alpha VoCs has been found by Fung et al. (2021) to be equal to 17.1% (95%CI 13-21%). According to Singanayagam et al. (2021), the SAR was found to be 38% for delta variant (with 95%CI 24-53%) in case of unvaccinated people, and 25% (with 95%CI 18-33%) for vaccinated people. The SAR for omicron is respectively 1.2 for vaccinated and 2.6 times more important than for delta (Lyngse et al., 2021).

Table 2.5: Secondary attack rates in household for different variants of concern and vaccination statuses.

VoC	Unvaccinated	X95 CI range	Vaccinated	X95 CI range 1	Multiplication factor toward a reference	Reference
original	17%	[14%-21%]				(Fung et al., 2021)
alpha	17%	[14%-21%]				(Fung et al., 2021)
delta	38%	[24%-53%]	25%	[18%-33%]		(Singanayagam et al., 2021)
omicron	57%		37.5%		1.5 more than delta	(Lyngse et al., 2021)

2.3.1.2.5 Infection probability

The possible triggering of the first infection phase for each worker was modeled using a dose-response modelling approach, considering the viral quantity inhaled by each one. Among different dose-response models proposed in the literature (see more details in the section 3.1 ), the Watanabe model (Watanabe, Bartrand, Weir, Omura, & Haas, 2010) was chosen herein to describe, for each worker, the probability to get infected as function of the number of inhaled virions (infectious virus), denoted \(dose\), as follows:

\[ P(infection) = 1 - \exp(-r * dose) \]

where \(r\) is the shape parameter defining the exponential form of the no-threshold model is a no-threshold model. The value of \(r\), estimated at 0.00246 [95%CI 0.00128-0.00527] based on literature data (cf. section 3.1) was used herein in the ABM model. It is also worth noting that the concentration of SARS-CoV-2 expressed in gene copies (e.g. RNA copies) or in infectious virus are strongly different. Important ratios between these concentrations have been suggested in the literature, varying from a factor of 100 to 1000 between the number of genes copies and the infectious viral particles (Miller et al., 2021; Pitol & Julian, 2021). This factor was included in the ABM model with a arbitrarily chosen value of 500 as well as the 95%CI between 100 and 1000.

2.3.1.3 Air

2.3.1.3.1 Exhalation

The initial source of SARS-CoV-2 was considered through different respiratory activities of the infectious workers emitting droplets of various sizes. The number of droplets (expressed as \(s^{-1}\) ) (containing or not viral particles) emitted over time was modeled as follows :

\[ \dot N_{exh, d} = \dot Q\upsilon_{exh} * C_d * (1-eff_m), \]

where \(\dot Q\upsilon_{exh}\) (\(m^3.h^{-1}\)) corresponds to the exhaled volumetric air flow rate, \(C_d\) (\(\#.m^{-3}\) ) is the concentration of droplets of diameter \(d\) and \(eff_m\) corresponds to the efficacy of the mask if correctly worn (otherwise, $eff_m = 0 $). The correct mask wearing was described using a parameter called the mask acceptability, denoted \(p_{mask}\), corresponding to the proportion of the workers constantly wearing a mask during their working day. Several types of mask can be included as input of the model such as surgical mask or FFP2, with their respective efficacy and acceptability.

The volumetric airflow rate was assumed to vary between the different workers depending on their physiology and the intensity of their activities. Based on data obtained by Buonanno, Morawska, & Stabile (2020), the ABM model considered six size classes for the droplets with the midpoint diameters \(d\) at 0.8, 1.8, 3.5, 5.5, 20 and 100 \(\mu m\), respectively. Several respiratory activities (events) were taken into account in the model, including breathing, talking, talking loud (based on study by Morawska et al. (2009)), coughing and sneezing (SARS-CoV-2 symptoms). The concentrations \(C_d\) associated with each droplet class emitted during each respiratory activity (event) estimated from studies by Morawska et al. (2009), Kennedy2009, Duguid1946 are gathered below:

Droplet concentrations during different respiratory activities

Respiratory activity	d = 0.8 \(\mu m\)	d = 1.8 \(\mu m\)	d = 3.5 \(\mu m\)	d = 5.5 \(\mu m\)	d = 20 \(\mu m\)	d = 100 \(\mu m\)
breathing	\(0.084 * 10^6\)	\(0.009* 10^6\)	\(0.003 * 10^6\)	\(0.002 * 10^6\)	\(0\)	\(0\)
talking	\(0.236 * 10^6\)	\(0.068 * 10^6\)	\(0.007 * 10^6\)	\(0.011 * 10^6\)	\(0\)	\(0\)
talking loud	\(0.751 * 10^6\)	\(0.139 * 10^6\)	\(0.139 * 10^6\)	\(0.059 * 10^6\)	\(0\)	\(0\)
coughing	\(0\)	\(0.025 * 10^6\)	\(0.145 * 10^6\)	\(0.48 * 10^6\)	\(2.51 * 10^6\)	\(6.65 * 10^6\)
sneezing	\(0\)	\(13 * 10^6\)	\(80 * 10^6\)	\(175 * 10^6\)	\(230 * 10^6\)	\(489 * 10^6\)

The occurrence frequency of the above respiratory activities during a time step varied from one worker to another, depending on their working activities as well as their infection phases. For example, Birring2006 suggested average coughing frequencies at 2 coughs per hour for healthy patients and 47 coughs per hour for symptomatic patients. Such variability between infection statuses was also suggested for the sneezing frequencies (Hansen&Mygind2002, Musch2021). These SARS-CoV-2 symptoms frequencies (expressed in \(\#.min^{-1}\)) included in the model are gathered below:

Symptoms	Infectious presymptomatic	Asymptomatic	Symptomatic
Coughing	0.033	0.033	0.783
Sneezing	0.002	0.08	0.017

Frequencies of different SARS-CoV-2 symptoms (expressed as the number of occurrences per minutes).

2.3.1.3.2 Droplet contamination

2.3.1.3.3 Aerosolization criteria

The different droplets emitted by the workers can be aerosolized (suspended in the air) (Bazant & Bush, 2021) or fall directly onto the nearby environments (including nearby workers, inert surfaces or food products) . This possible aerosolization was determined using the

2.3.1.3.4 Sedimentation

2.3.1.3.5 Falling & Projection

Figure 2.6: Distribution of the falling and projected droplets onto nearby environments and workers

2.3.1.3.6 Inhalation

2.3.1.4 Surfaces

inert surfaces

2.3.1.5 Food

2.3.1.5.1 Characteristics

• The type of meat
• Ncarcass_daily
• carcass_weight
• net_weight
• CSU

2.3.1.5.2 Rules

• All characteristics of the agents “food portions” were initialized daily. The food portions came from different carcasses. The total number of carcasses (and agents) was calculated each day depending on the expected total meat quantity produced by the establishment (expressed as number of carcasses per day or per week). Each initialized agent had a unique ID (e.g: B_12_028_004 corresponds to the 4th portion of the 28th carcass of the 12th day, “B” stands for “bovine”).

• The duration (expressed in minutes) corresponding to different cutting steps for each carcass and for separating food portions at different locations in the plant were estimated/assumed based on information from industrial interviews. Hence, knowing this duration, the carcasses could arrive sequentially and/or simultaneously in the cutting room depending on the expected number of carcasses per day.

2.3.1.5.3 Transfers by contact

Figure 2.7: Tranfers of the droplets between the different inert surfaces and food portions

2.3.2 Input data

2.3.2.1 Reference scenario

Scenario chosen based on an epidemiological investigation on several SARS-CoV-2 clusters in France during the period between May 2020 and April 2021.

For each simulation, the model was initialized at the first day with an initial number of infected workers \(n_{Contaminated-Init} = 1\)

2.3.2.2 Model parameter values

2.4 Parallel computing

Calculation / technical requirements / Migale

2.5 Convergence check

approach for convergence /

3 Appendices

3.1 Dose-response model parameters

Dose-response modelling is the process of using mathematical relationships to describe the probability of an adverse health effect (e.g., infection, illness) occurring in an individual or the frequency of an adverse health effect in a population when that individual or population is exposed to a specific dose of pathogenic microorganisms. Two types of modelling approaches can be distinguished (Haas, 2002). In the first, it is considered that there is a cooperation between ingested/inhaled microorganisms, involving the notion of minimum infective dose: the infection is the result of the combined action of multiple micro-organism units, and it becomes possible only if the dose exceeds a threshold (the so-called “minimal infective dose”). In the second approach it is assumed on an independence of action of microorganism units. Each microorganism unit has a non-zero chance of causing infection on its own. For SARS-CoV-2, as for other infectious pathogens, the models currently used are based on the single hit model theory and the absence of a threshold (Poydenot et al., 2021).

• Exponential dose response model

The exponential model is a no-threshold model, where the probability of infection (r) for a virion particle is constant, and the number of virions inhaled is not perfectly known but assumed to follow a Poisson distribution of parameter d . The hypothesis of the constancy of r means that the result of the interaction between host, microorganism and environmental factors does not vary much between exposed individuals. The Poisson distribution is applicable in case the contamination is considered homogeneous. This assumption appears suitable for characterizing droplet distribution in the air. The probability of being infected following the inhalation of air in which the average contamination is equal to d is expressed as follows:

\[P(\mathrm{infection}|d)=1-\exp(-r/d)\]

Figure 3.1: Examples of the exponential dose-response model for calculating the probability opf infection according to the dose for different r values**

• Wells-Riley model

The Wells-Riley model can also be used to calculate the probability of infection based on given dose (Sze To & Chao, 2010; Wells et al., 1955). It also relies on the hypothesis of a random distribution of the pathogen in the air. But it uses another parametrisation compared to the exponential dose-response model. The quantum of infection q is used. q corresponds to a quantity of inhaled virions for which the probability of infection in the population is 63%.

\[P(\mathrm{infection}|d)=1-\exp(-d/q)\]

Figure 3.2: Illustration of the notion of quantum of infection. q = 100 in this example

• Comparison of models

No data have been used so far for fitting an exponential dose response curve for Sars-Cov2 (Parhizkar, Van Den Wymelenberg, Haas, & Corsi, 2021). However prior work has shown that dose-responses for other coronaviruses (Haas, 2021; Watanabe et al., 2010). This dose-response has been used in several risk assessment for Sars-Cov2 (e.g. Pitol & Julian, 2021).

Prentiss, Chu, & Berggren (2020) provided some estimates of q values based on estimation of exposure and observed attack rates. Six values were obtained from 6 six different situations were the attack rates. The values ranged from 322 to 3012. Augenbraun et al. (2020) assumed a conservative value of q equal to 100.

Figure 3.3 compared the exponential dose response derived from Watanabe et al. (2010) and available in QMRAwiki and the values derived from Prentiss et al. (2020).

Figure 3.3: Comparison of different dose-response models

The Watanabe’s model provided a conservative estimate of the infection probability. The best fit value for \(r = 0.00246\) was used in the SACADA_ABM model. The 95% CI are used in the sensitivity analysis of the model (0.00128 and 0.00527).

• Expression of dose to assess probability of infection

Evidence suggests that the concentration of SARS-Cov2 expressed in gene copies or in infectious virus are strongly different. Different ratios have been proposed to account of this difference. Infective virus are though to be from 100 to 1000 lower than gene copies (Miller et al., 2021; Pitol & Julian, 2021). The value of 500 is used in SACADA_ABM. The values of 100 to 1000 will be used in the sensitivity analysis.

References

Alene, M., Yismaw, L., Assemie, M. A., Ketema, D. B., Mengist, B., Kassie, B., & Birhan, T. Y. (2021). Magnitude of asymptomatic COVID-19 cases throughout the course of infection: A systematic review and meta-analysis. PloS One, 16(3), e0249090.

Augenbraun, B. L., Lasner, Z. D., Mitra, D., Prabhu, S., Raval, S., Sawaoka, H., & Doyle, J. M. (2020). Assessment and mitigation of aerosol airborne SARS-CoV-2 transmission in laboratory and office environments. Journal of Occupational and Environmental Hygiene, 17(10), 447–456.

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